John Pfaff is a professor of law at Fordham University. He is the author of “Locked In: The True Causes of Mass Incarceration and How to Achieve Real Reform.”

Criminal justice policy has secured a prominent place in the Democratic presidential primary race, with several candidates offering plans to reform the way Americans punish. Reinventing how we handle crime is one of the most important civil rights challenges of our time, yet much of the debate about criminal justice is built on misconceptions that often push reform in inadequate and even counterproductive directions.

Myth No. 1: U.S. prisons are full of nonviolent drug offenders.

Asked recently about voting rights for felons, Sen. Cory Booker (N.J.), one of the Democratic presidential candidates, claimed that “we locked up more people for marijuana in 2017 than all the violent crimes combined.” Rep. Alexandria Ocasio-Cortez (D-N.Y.) has echoed that view, suggesting on Twitter that the prison system is defined by nonviolent people “stopped w/ a dime bag.”

But the simple truth is that, at a minimum, 55 percent of those in state prison have been convicted of a violent crime — and more than half of these people, or nearly 30 percent of the total prison population, have been found guilty of murder, manslaughter, rape or sexual assault, according to the Bureau of Justice Statistics. Slightly less than 15 percent are incarcerated for drug crimes, even though most Americans believe the figure to be about 50 percent. (Drugs play a bigger role in the federal prison system, but that holds only about 10 percent of all prisoners; most incarcerated people are in state prison.)

The share of those in state prison for committing violence is even greater than 55 percent, however. Prisoners are classified by the most serious offense for which they are convicted, not arrested or charged. So if someone is arrested for a violent crime but ends up pleading guilty to a drug charge, his crime is classified as a nonviolent drug offense, even if the underlying incident - like a domestic violence case in which the victim won’t testify - is the reason the prosecutor sought prison time.

Myth No. 2: Private prisons drive mass incarceration.

When people try to explain how the United States ended up with nearly 25 percent of the world’s prisoners, they often point to firms that directly profit from incarceration by running prisons or by providing services to public facilities. At a recent presidential campaign event, Sen. Elizabeth Warren (D-Mass.) blamed private prisons for mass incarceration (“We need to get rid of for-profit, private prisons”). Sen. Bernie Sanders (I-Vt.) focused on private prisons in his 2016 presidential bid and is doing so again (“The private prison racket has got to end”).

There are two central flaws in this claim. First, only about 8 percent of all state and federal prisoners are held in private facilities . Most of those in private prisons are held in just five states, and there is no real evidence that prison populations have grown faster in those states than elsewhere.

Second, of the roughly $50 billion we spend on prisons, about two-thirds , or $30 billion, is spent on wages and benefits for public-sector employees. In comparison, private prison firms collectively earn a few billion in revenue and (more important for their incentives to lobby) about $300 million in profits — just 1 percent of the public-sector wage bill. So public-sector correctional officer unions have a reason to lobby against reforms that would reduce inmate populations, especially since prisons often provide some of the only well-paying jobs in the rural communities where they are located.

The reality is more complicated. Among prisoners who are over 65, fully half have served fewer than 10 years, which means they were admitted in their mid- to late 50s. Much of the aging of the prison population plausibly comes not from long sentences but from our inability to address late-age violence properly. Age-related illnesses like Alzheimer’s and frontotemporal dementia can lead to violent behavior, and far too often we rely on police and jails, not public health services, to manage it. One study also suggests that extensive drug use when younger has led to higher levels of crime among a cohort of older Americans.

There is a problem with how we think about the other half of older inmates as well, those who were admitted when younger and are growing old behind bars. Even as we increasingly understand the problems and costs with people aging in prison, we often restrict reform efforts aimed at them, such as compassionate-release programs, to those convicted of nonviolent crimes - even though nearly 90 percent are serving time for violent crimes, frequently murder or manslaughter.

Perhaps the single most important statistic for evaluating new criminal justice policies is their effect on recidivism. So it would be great if recidivism statistics actually measured reoffending. They don’t. Formal recidivism numbers measure something like “the number of people rearrested” or “the number of people readmitted to prison.” Such data both understates and overstates actual reoffending, glossing over a critical debate about what recidivism even means.

First, the undercounting: We can't measure reoffending that no one detects. No one voluntarily reports crimes they committed that the police did not observe or that victims did not report. This can exacerbate racial disparities, too: Considering the communities most heavily policed, reoffending by whites is much less likely to be detected than that by blacks.

The overcounting issue is less immediately obvious but arguably far more important. If someone shifts from committing one robbery per day to one per month, he could eventually be reconvicted and classified as a recidivist. That the amount of harm he caused fell by about 95 percent is immaterial. Ours is a binary metric; any one failure counts as recidivism. It's a metric inconsistent with what we know about the rocky reality of desistance.

Myth No. 5: Not sending someone to prison saves about $35,000 a year.

Criminal justice reform is one of the few bipartisan issues in politics these days, and a key goal of the conservative push for change is to reduce or reallocate government spending. It’s common to see claims that for each person kept out of prison, taxpayers will save about $35,000 a year. The Brennan Center for Justice, for example, argued that cutting prison populations by 576,000 would save about $20 billion annually, and a 2018 report from the White House Council of Economic Advisers on the fiscal impact of reducing recidivism relied on a roughly similar estimate.

It’s easy to see where the number comes from: We spend about $50 billion to lock up 1.4 million people per year in state prisons. That’s about $35,000 per person in prison per year. Yet that is not what we save when we release a single inmate from prison.

Recall that about two-thirds of prison spending is wages: If we don’t cut payroll in proportion to inmate population, then the savings per prisoner will be much less. Other fixed costs, like water and heating bills, likewise do not decline steadily with the number of prisoners. The real savings of reducing the prison population by one person is often about $4,000 to $16,000, not $35,000.

If we release enough people that we can close an entire prison wing or facility, perhaps we can save something closer to the estimated average. But states often keep mostly empty facilities open, and when they do close institutions, they often relocate most of the correctional officers to other facilities.